Properties

Label 4-9702e2-1.1-c1e2-0-32
Degree $4$
Conductor $94128804$
Sign $1$
Analytic cond. $6001.73$
Root an. cond. $8.80175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·4-s + 4·8-s + 2·11-s + 5·16-s + 4·22-s − 8·23-s − 8·25-s − 12·29-s + 6·32-s − 16·37-s − 16·43-s + 6·44-s − 16·46-s − 16·50-s − 24·58-s + 7·64-s − 16·67-s − 16·71-s − 32·74-s + 8·79-s − 32·86-s + 8·88-s − 24·92-s − 24·100-s + 16·107-s + 8·109-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s + 0.603·11-s + 5/4·16-s + 0.852·22-s − 1.66·23-s − 8/5·25-s − 2.22·29-s + 1.06·32-s − 2.63·37-s − 2.43·43-s + 0.904·44-s − 2.35·46-s − 2.26·50-s − 3.15·58-s + 7/8·64-s − 1.95·67-s − 1.89·71-s − 3.71·74-s + 0.900·79-s − 3.45·86-s + 0.852·88-s − 2.50·92-s − 2.39·100-s + 1.54·107-s + 0.766·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(94128804\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(6001.73\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9702} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 94128804,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 24 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2^2$ \( 1 + 72 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 80 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 48 T^{2} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.32473626253325216386573207573, −7.19576533044290010539002298808, −6.68892702580297610804645332557, −6.40238640090810746421784041912, −6.05560860576785897999365049570, −5.77536415636543509426383610308, −5.36923663389036531189594286452, −5.27266830018561130165696431117, −4.62196015954846721481784902781, −4.40729540099479425105988438856, −3.88447990559653417784001258306, −3.74389898343624420507283980275, −3.24542672905485775262595585305, −3.22075544445163111007606497831, −2.30560060491952324788022815643, −1.94697167838855427000488868940, −1.74846006970111332048372191005, −1.36271820470871749792875197880, 0, 0, 1.36271820470871749792875197880, 1.74846006970111332048372191005, 1.94697167838855427000488868940, 2.30560060491952324788022815643, 3.22075544445163111007606497831, 3.24542672905485775262595585305, 3.74389898343624420507283980275, 3.88447990559653417784001258306, 4.40729540099479425105988438856, 4.62196015954846721481784902781, 5.27266830018561130165696431117, 5.36923663389036531189594286452, 5.77536415636543509426383610308, 6.05560860576785897999365049570, 6.40238640090810746421784041912, 6.68892702580297610804645332557, 7.19576533044290010539002298808, 7.32473626253325216386573207573

Graph of the $Z$-function along the critical line